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62 views

finite volume for diffusion equation with anisotropic (tensor) coefficient

Consider the scalar PDE for $u$ with Dirichlet boundary conditions: $\mathrm{div}(\mathcal{K}\nabla u) = f\; \forall x\; \in \Omega \subset R^2$, $u = 0 \; \forall \; x\;\in \partial\Omega$ ...
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0answers
25 views

Interface Formulation at Finite Volume Boundaries when using the Dual Mesh

When using the dual mesh (vertex-centered) for finite volume methods, you end up with a cell center at the boundaries between materials. It is possible that the equations being solved in each ...
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0answers
36 views

data structures for efficient/easy implementation of finite volume method for 2D Poisson equation

My question is about implementation alone. Consider a square domain with regular square, cell centred finite volumes. This is for the multiscale finite volume method (Jenny and Lunati) I need to ...
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0answers
54 views

Assembling sparse matrix in PETSC for Poisson equation

I am a novice at PETSC, and I have been trying to write an FVM code for steady heat conduction in 2D using PETSC (square, regular grid, Dirichlet boundaries) Since the large matrix , say A, will be ...
3
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2answers
172 views

Developing finite volume (FVM) code in C . General advice

I need to develop a FVM code in C (The multiscale FVM method for heterogeneous media). I know that: Only uniform rectangular grids will be considered (2d now, later 3d) Sparse systems will be large ...
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1answer
122 views

Euler's equations 1d for pipe, Inlet boundary conditions

$\def\rmin{{\mathrm{in}}}$ $\def\l{\left}\def\r{\right}$ $\def\tagl#1{\tag{#1}\label{#1}}$ I am using the one-dimensional finite volume method to calculate the air flow in some tube. For subsonic ...
3
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1answer
50 views

Conservative FV Immersed boundary method for compressible flow

Is there a conservative FV second-order (or first-order) accurate immersed boundary method for compressible flow including moving boundaries (in the literature)? By compressible flow I mean the ...
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0answers
52 views

Implementation of convection scheme given by normalized variable diagram

In finite difference and finite volume methods, convection schemes (upwind, central, quick, ...) are usually shown in a normalized variable diagram. The diagram gives the normalized face variable as a ...
3
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0answers
75 views

Finite volume method

I have question connected with finite volume method. Consider equation $$\frac{\partial u}{\partial t}=\operatorname{div}A\nabla u +f . \quad x\in \overline{B}_{1} (0)\subset \mathbb{R}^3 -\text{unit ...
1
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1answer
156 views

Reconstructing fluxes

Given a standard advection equation, we write the update as $$ q_i^{n+1}=q_i^n+\frac{\Delta t}{\Delta x}\left(F_i^{n+1/2}-F_{i+1}^{n+1/2}\right) $$ with $F_i^{n+1/2}=F\left(q_i^n,\,q_i^*\right)$ and ...
5
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1answer
127 views

Finite Volume Implementation

I am trying to implement a simple finite volume method solver. I had a class on FVM a while back, but am still aware of the principal concepts. But implementing the FVM for non-cartesian or 1D meshes ...
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0answers
52 views

Characteristic length of differential element of cylinder surface?

I am trying to find the Nusselt number for a small element of the outside of a cylinder that has a height of $\Delta z$. I found the average Grashof number of a surface as $$Gr_{L}=\frac{\beta \rho ...
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0answers
79 views

Problem with cell size and boundary conditions in transient cylindrical conduction

I am attempting to model the steady state behavior of a cylinder using the finite volume method (FVM) subjected to a variety of boundary conditions in Matlab. First off, I am treating the cylinder as ...
5
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0answers
126 views

Interface Conductivity for Finite Volume Method Heat Transfer in Cylindrical Coordinates

I'm solving a heat conduction problem in cylindrical coordinates with a composite cylinder made of two different materials. Essentially the cylinder is split into a central cylinder of material A, ...
4
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1answer
103 views

Is is possible to mix finite-volume and finite-difference discretisations when solving a coupled non-linear system?

In my last question I noticed a peculiar error when solving the Poisson equation using finite volume method (FVM), Peculiar error when solving the Poisson equation on a non-uniform mesh (1D only) ...
6
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2answers
159 views

Is there a good tutorial or textbook-like source on implementing ENO/WENO with limiters in one (and more than one) dimension?

I've inherited a finite volume code that does a second-order discretization of flux terms for a set of mixed parabolic-elliptic equations with discontinuous diffusion coefficients. The impression I ...
4
votes
1answer
90 views

How can exponential fitting be used with the finite element method?

Restricted to one dimensional problem, is it possible to dynamically adapt the finite element method (FEM) discretisation based on the local value of the P├ęclet number ($P_e$) for advection-diffusion ...
4
votes
2answers
203 views

Does the finite element method impose any restrictions on the Peclet number for numerical stability?

Background on finite volume method When discretising the flux with a central difference stencil of the the advection-diffusion equation restriction $\frac{ah}{d} < 2$ must be observed for the ...
5
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2answers
210 views

Peculiar error when solving the Poisson equation on a non-uniform mesh (1D only) finite volume method

I have been trying to debug this error the last few days I wondered if anybody has advice on how to proceed. I am solving the Poisson equation for a step charge distribution (a common problem in ...
6
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3answers
425 views

Applying Dirichlet boundary conditions to the Poisson equation with finite volume method

I would like to know how Dirichlet conditions are normally applied when using the finite volume method on a cell-centered non-uniform grid, My current implementation simply imposes the boundary ...
4
votes
1answer
224 views

A simple function for generating a nonuniform mesh in 1D with fixed minimum spacing

I am solving an advection-diffusion problem where the solution variable is mostly flat apart from a small region near the centre of the domain where there are shape gradients. I would like to generate ...
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3answers
3k views

What are the conceptual differences between the finite element and finite volume method?

There is an obvious difference between finite difference and the finite volume method (moving from point definition of the equations to integral averages over cells). But I find FEM and FVM to be very ...
5
votes
2answers
132 views

Solving PDE with spatial and temporal derivatives on left hand side

I wish to solve an equation of the form, $$ \frac{\partial}{\partial t} \left( \frac{\partial \phi}{\partial x} \right) = -\frac{\partial}{\partial x}(\mathcal{F}) $$ for the variable $\phi$ (e.g. ...
1
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1answer
157 views

How to write this non-linear PDE with the finite volume method?

I wish to solve a coupled system of non-linear equation of this form, $$ u_t = -(\mathcal{F})_x + f(x,u,w) \\ w_t = \mathcal{F} + g(x,u,w) $$ by stepping the equations forward in time. The first ...
9
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2answers
234 views

Data structures for finite volume code: Arrays vs Classes

I have to write a finite volume code for Magnetohydrodynamics (MHD). I have written numerical code before but not at this scale. I just wanted to ask which will be a good choice, using a data ...
5
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2answers
279 views

Finite-volume method: can Dirichlet boundary conditions be applied to the integral form?

I would like to apply Dirichlet conditions to the advection-diffusion equation using the finite-volume method. This answer, "How should boundary conditions be applied when using finite-volume method?" ...
2
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1answer
74 views

Finite volume with cell averages vs cell totals for conservation equations

What implementation details need to change if I use a cell average approach rather than a cell total approach for the finite-volume method? For example, consider the conservation law, $$ u_t + ...
13
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1answer
1k views

How should boundary conditions be applied when using finite-volume method?

Following from my previous question I am trying to apply boundary conditions to this non-uniform finite volume mesh, I would like to apply a Robin type boundary condition to the l.h.s. of the ...
5
votes
1answer
183 views

Are we free to choose the position of ghost cells on a non-uniform finite-volume mesh?

Following Hundsdorfer approach the finite volume discretisation of the advection-diffusion equation (conservative form) on non-uniform cell centered grid can be written as, $$ w_j^{\prime} = ...
9
votes
3answers
231 views

How should non-constant coefficients be treated with finite-volume first order upwind scheme?

Starting with the advection equation in conservation form. $$ u_t = (a(x)u)_x $$ where $a(x)$ is a velocity which depend on space, and $u$ is a concentration of a species which is conserved. ...
2
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1answer
46 views

Effect of Normalization in Unknowns

When solving a FV formulation of a set of equations, a code I am currently working with has user defined normalization factors for scaling equations. It normalizes time, number densities, potential, ...
10
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2answers
209 views

Connections between Differential Forms and the second order Finite Volume Method

Reading today about the theory of differential forms, I was left impressed how much it reminded me of second order Finite Volume Method (FVM). I'm struggling to figure out is thinking this way just ...
8
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1answer
951 views

OpenFoam vs FiPy

I need to learn and utilize a finite volume automated solution package for a project I'm working on and have narrowed it down to these two packages. I was wondering if anybody has experience of both ...
4
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1answer
523 views

How to approximate flux (with gradient) when using finite volumes?

In finite volume method one is using cell averages. In nonlinear conservation laws discontinuities can be created in the solution process. How to compute the flux when the flux contains a gradient ...
5
votes
2answers
149 views

How to do upwinding in finite volume schemes for nonlinear equations?

In finite difference theory, you learn, that you have to use upwinding for equations with high convection, like Burgers' equation. What does the finite volume equivalent look like? What if the ...
9
votes
1answer
171 views

Hybrid spatial schemes for CFD: any downside to blending versus switching?

Aside from extra computational cost due to having to compute both fluxes over a certain region, is there any downside to blend two flux evaluations for a hybrid scheme in a finite volume method? The ...
9
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2answers
309 views

What are some good data-types for unstructured cell-centered FVM CFD code?

I'm interested in an advice for efficient data structures for cell browsing in unstructured cell-based finite volume CFD. One example that I encountered (in dolfyn cfd code) goes like this (I'll show ...
4
votes
1answer
727 views

Finite Volume Method vs. Finite Element Method for Eulerian and Lagrangian Reference Frames

As far as I can tell, finite volume methods are primarily for eulerian reference frames (such as in fluid dynamics simulations) while finite element methods are easily ameanable to lagrangian ...
5
votes
1answer
106 views

a priori error analysis of cell-centered finite-volume methods

I'm using the cell-centered finite-volume method (for example Morton, Numerical solution of convection-diffusion problems, Chapman&Hall, 1996) to discretize the advection-diffusion equation and ...
5
votes
1answer
494 views

PDE discretization with the method of rothe and the method of lines (Modular implementation)

The Heat equation is discretized in space with FV (or FEM), and a semi-discrete equation is obtained (system of ODEs). This approach, known as the method of lines, allows to easily switch from one ...
5
votes
2answers
243 views

Interpolation schemes to move data between cells and nodes

I work on non-graded quadtree grids where the entire grid is a hierarchy of cells specified using a quadtree data structure, where, in general, there is no constraint regarding the relative size of ...
2
votes
2answers
284 views

Finite volume solution of electrostatics using magnetic vector potential

I would like to solve for the electric potential and magnetic vector potential using the finite volume method (collocated grid). My equations are: $\nabla\cdot(\sigma\nabla\phi)=0$ $\nabla \cdot ...
3
votes
1answer
140 views

High-resolution finite volume schemes for two phase flow (fields with jumps) literature sources

what other recent sources of literature on this topic would you recommend? This is where I'm starting from: Leveque's article: HRIC scheme But the related articles seem to be a bit dated (some up ...